What Is a Tessellation?

Through a dialogue with Michael South, then at the Shodor Organization, and Sarah Seastone, at the Math Forum, I have thought more about what a tessellation actually is. It seems that of the six examples of tessellations that I have presented, only two of them appear at first to really qualify as tessellations! With some work and/or dialogue perhaps the other four examples can be justified or altered a bit to qualify as tessellations.

Michael South wrote to me with the following thoughts about whether my HyperCard and HyperStudio exercises are tessellations.

"I think most definitions of tessellation require that the figure cover the plane, with no spaces left. The entire boundary of whatever your basic 'tiling unit' is fits right next to the boundaries of the adjoining tiles. The first example of the person with the hat (both in HyperCard and HyperStudio) would be a tessellation in this sense, but the other two (the squiggles and the triangles) would not.

"Suppose you had a square frame of such a dimension that pool balls fit into it exactly in a 4X4 square. Now consider that as a 2-D picture viewed from the top - a four by four 'grid' of circles."

"Now, technically, those circles don't tessellate the plane - they leave gaps. But they sort of quasi-tessellate by suggesting a regular pattern (say, of squares with circles inscribed) that do tessellate.

"Or, in a regular triangle from a pool table, the arrangement of the balls suggests a hexagon pattern which tessellates, although the balls, again, do not."

"There is some shape suggested by the squiggle pattern that tessellates the plane, and if you did tile the plane with it, with each tile decorated with a squiggle pattern, and then erased the boundary lines of the tiles, you would get the pattern you have. It makes an interesting question; in a sense it is richer than most methods of creating tessellation patterns in that it allows one to wonder what the exact tessellating tile is. You might be able to call them 'hidden edge tessellations' and then explore the question of how one would find/define the edge."